On an invariant measure for homeomorphisms of a circle with a point of break
The authors study homeomorphisms of the circle \(S^1=[0,1)\) with a simple point of break, that is, with a jump of the first derivative at a point. As is well known in the case of generic rotation numbers, the invariant measure for a \(C^{2+\nu}\)-diffeomorphism is absolutely continuous with respect to the Lebesgue measure. The authors show that the situation becomes diametrically opposite in the presence of a single point of break-type singularity, namely, they prove that an invariant measure is always singular with respect to the Lebesgue measure.
scientific article; zbMATH DE number 1292496